fact stringlengths 6 3.84k | type stringclasses 11 values | library stringclasses 32 values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
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ContinuousAt.comp_div_cases {f g : α → G₀} (h : α → G₀ → β) (hf : ContinuousAt f a)
(hg : ContinuousAt g a) (hh : g a ≠ 0 → ContinuousAt ↿h (a, f a / g a))
(h2h : g a = 0 → Tendsto ↿h (𝓝 a ×ˢ ⊤) (𝓝 (h a 0))) :
ContinuousAt (fun x => h x (f x / g x)) a := by
change ContinuousAt (↿h ∘ fun x => (x, f x / g x)) a
by_cases hga : g a = 0
· rw [ContinuousAt]
simp_rw [comp_apply, hga, div_zero]
exact (h2h hga).comp (continuousAt_id.tendsto.prodMk tendsto_top)
· fun_prop (disch := assumption) | theorem | Topology | [
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.GroupWithZero.Units.Equiv",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Homeomorph.Lemmas"
] | Mathlib/Topology/Algebra/GroupWithZero.lean | ContinuousAt.comp_div_cases | Continuity at a point of the result of dividing two functions continuous at that point, where
the denominator is nonzero. -/
nonrec theorem ContinuousAt.div (hf : ContinuousAt f a) (hg : ContinuousAt g a) (h₀ : g a ≠ 0) :
ContinuousAt (f / g) a :=
hf.div hg h₀
@[continuity]
theorem Continuous.div (hf : Continuous f) (hg : Continuous g) (h₀ : ∀ x, g x ≠ 0) :
Continuous (f / g) := by simpa only [div_eq_mul_inv] using hf.mul (hg.inv₀ h₀)
theorem continuousOn_div : ContinuousOn (fun p : G₀ × G₀ => p.1 / p.2) { p | p.2 ≠ 0 } :=
continuousOn_fst.div continuousOn_snd fun _ => id
@[fun_prop]
theorem Continuous.div₀ (hf : Continuous f) (hg : Continuous g) (h₀ : ∀ x, g x ≠ 0) :
Continuous (fun x => f x / g x) := by
simpa only [div_eq_mul_inv] using hf.mul (hg.inv₀ h₀)
@[fun_prop]
theorem ContinuousAt.div₀ (hf : ContinuousAt f a) (hg : ContinuousAt g a) (h₀ : g a ≠ 0) :
ContinuousAt (fun x => f x / g x) a := ContinuousAt.div hf hg h₀
@[fun_prop]
theorem ContinuousOn.div₀ (hf : ContinuousOn f s) (hg : ContinuousOn g s) (h₀ : ∀ x ∈ s, g x ≠ 0) :
ContinuousOn (fun x => f x / g x) s := ContinuousOn.div hf hg h₀
/-- The function `f x / g x` is discontinuous when `g x = 0`. However, under appropriate
conditions, `h x (f x / g x)` is still continuous. The condition is that if `g a = 0` then `h x y`
must tend to `h a 0` when `x` tends to `a`, with no information about `y`. This is represented by
the `⊤` filter. Note: `tendsto_prod_top_iff` characterizes this convergence in uniform spaces. See
also `Filter.prod_top` and `Filter.mem_prod_top`. |
Continuous.comp_div_cases {f g : α → G₀} (h : α → G₀ → β) (hf : Continuous f)
(hg : Continuous g) (hh : ∀ a, g a ≠ 0 → ContinuousAt ↿h (a, f a / g a))
(h2h : ∀ a, g a = 0 → Tendsto ↿h (𝓝 a ×ˢ ⊤) (𝓝 (h a 0))) :
Continuous fun x => h x (f x / g x) :=
continuous_iff_continuousAt.mpr fun a =>
hf.continuousAt.comp_div_cases _ hg.continuousAt (hh a) (h2h a) | theorem | Topology | [
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.GroupWithZero.Units.Equiv",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Homeomorph.Lemmas"
] | Mathlib/Topology/Algebra/GroupWithZero.lean | Continuous.comp_div_cases | `h x (f x / g x)` is continuous under certain conditions, even if the denominator is sometimes
`0`. See docstring of `ContinuousAt.comp_div_cases`. |
protected mulLeft₀ (c : α) (hc : c ≠ 0) : α ≃ₜ α :=
{ Equiv.mulLeft₀ c hc with
continuous_toFun := continuous_mul_left _
continuous_invFun := continuous_mul_left _ } | def | Topology | [
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.GroupWithZero.Units.Equiv",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Homeomorph.Lemmas"
] | Mathlib/Topology/Algebra/GroupWithZero.lean | mulLeft₀ | Left multiplication by a nonzero element in a `GroupWithZero` with continuous multiplication
is a homeomorphism of the underlying type. |
protected mulRight₀ (c : α) (hc : c ≠ 0) : α ≃ₜ α :=
{ Equiv.mulRight₀ c hc with
continuous_toFun := continuous_mul_right _
continuous_invFun := continuous_mul_right _ }
@[simp] | def | Topology | [
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.GroupWithZero.Units.Equiv",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Homeomorph.Lemmas"
] | Mathlib/Topology/Algebra/GroupWithZero.lean | mulRight₀ | Right multiplication by a nonzero element in a `GroupWithZero` with continuous multiplication
is a homeomorphism of the underlying type. |
coe_mulLeft₀ (c : α) (hc : c ≠ 0) : ⇑(Homeomorph.mulLeft₀ c hc) = (c * ·) :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.GroupWithZero.Units.Equiv",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Homeomorph.Lemmas"
] | Mathlib/Topology/Algebra/GroupWithZero.lean | coe_mulLeft₀ | null |
mulLeft₀_symm_apply (c : α) (hc : c ≠ 0) :
((Homeomorph.mulLeft₀ c hc).symm : α → α) = (c⁻¹ * ·) :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.GroupWithZero.Units.Equiv",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Homeomorph.Lemmas"
] | Mathlib/Topology/Algebra/GroupWithZero.lean | mulLeft₀_symm_apply | null |
coe_mulRight₀ (c : α) (hc : c ≠ 0) : ⇑(Homeomorph.mulRight₀ c hc) = (· * c) :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.GroupWithZero.Units.Equiv",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Homeomorph.Lemmas"
] | Mathlib/Topology/Algebra/GroupWithZero.lean | coe_mulRight₀ | null |
mulRight₀_symm_apply (c : α) (hc : c ≠ 0) :
((Homeomorph.mulRight₀ c hc).symm : α → α) = (· * c⁻¹) :=
rfl | theorem | Topology | [
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.GroupWithZero.Units.Equiv",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Homeomorph.Lemmas"
] | Mathlib/Topology/Algebra/GroupWithZero.lean | mulRight₀_symm_apply | null |
map_mul_left_nhds₀ (ha : a ≠ 0) (b : G₀) : map (a * ·) (𝓝 b) = 𝓝 (a * b) :=
(Homeomorph.mulLeft₀ a ha).map_nhds_eq b | theorem | Topology | [
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.GroupWithZero.Units.Equiv",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Homeomorph.Lemmas"
] | Mathlib/Topology/Algebra/GroupWithZero.lean | map_mul_left_nhds₀ | null |
map_mul_left_nhds_one₀ (ha : a ≠ 0) : map (a * ·) (𝓝 1) = 𝓝 (a) := by
rw [map_mul_left_nhds₀ ha, mul_one] | theorem | Topology | [
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.GroupWithZero.Units.Equiv",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Homeomorph.Lemmas"
] | Mathlib/Topology/Algebra/GroupWithZero.lean | map_mul_left_nhds_one₀ | null |
map_mul_right_nhds₀ (ha : a ≠ 0) (b : G₀) : map (· * a) (𝓝 b) = 𝓝 (b * a) :=
(Homeomorph.mulRight₀ a ha).map_nhds_eq b | theorem | Topology | [
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.GroupWithZero.Units.Equiv",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Homeomorph.Lemmas"
] | Mathlib/Topology/Algebra/GroupWithZero.lean | map_mul_right_nhds₀ | null |
map_mul_right_nhds_one₀ (ha : a ≠ 0) : map (· * a) (𝓝 1) = 𝓝 (a) := by
rw [map_mul_right_nhds₀ ha, one_mul] | theorem | Topology | [
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.GroupWithZero.Units.Equiv",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Homeomorph.Lemmas"
] | Mathlib/Topology/Algebra/GroupWithZero.lean | map_mul_right_nhds_one₀ | null |
nhds_translation_mul_inv₀ (ha : a ≠ 0) : comap (· * a⁻¹) (𝓝 1) = 𝓝 a :=
((Homeomorph.mulRight₀ a ha).symm.comap_nhds_eq 1).trans <| by simp | theorem | Topology | [
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.GroupWithZero.Units.Equiv",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Homeomorph.Lemmas"
] | Mathlib/Topology/Algebra/GroupWithZero.lean | nhds_translation_mul_inv₀ | null |
ContinuousInv₀.of_nhds_one (h : Tendsto Inv.inv (𝓝 (1 : G₀)) (𝓝 1)) :
ContinuousInv₀ G₀ where
continuousAt_inv₀ x hx := by
have hx' := inv_ne_zero hx
rw [ContinuousAt, ← map_mul_left_nhds_one₀ hx, ← nhds_translation_mul_inv₀ hx',
tendsto_map'_iff, tendsto_comap_iff]
simpa only [Function.comp_def, mul_inv_rev, mul_inv_cancel_right₀ hx']
@[deprecated (since := "2025-09-01")] alias HasContinuousInv₀.of_nhds_one :=
ContinuousInv₀.of_nhds_one | theorem | Topology | [
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.GroupWithZero.Units.Equiv",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Homeomorph.Lemmas"
] | Mathlib/Topology/Algebra/GroupWithZero.lean | ContinuousInv₀.of_nhds_one | If a group with zero has continuous multiplication and `fun x ↦ x⁻¹` is continuous at one,
then it is continuous at any unit. |
continuousAt_zpow₀ (x : G₀) (m : ℤ) (h : x ≠ 0 ∨ 0 ≤ m) :
ContinuousAt (fun x => x ^ m) x := by
rcases m with m | m
· simpa only [Int.ofNat_eq_coe, zpow_natCast] using continuousAt_pow x m
· simp only [zpow_negSucc]
have hx : x ≠ 0 := h.resolve_right (Int.negSucc_lt_zero m).not_ge
exact (continuousAt_pow x (m + 1)).inv₀ (pow_ne_zero _ hx) | theorem | Topology | [
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.GroupWithZero.Units.Equiv",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Homeomorph.Lemmas"
] | Mathlib/Topology/Algebra/GroupWithZero.lean | continuousAt_zpow₀ | null |
continuousOn_zpow₀ (m : ℤ) : ContinuousOn (fun x : G₀ => x ^ m) {0}ᶜ := fun _x hx =>
(continuousAt_zpow₀ _ _ (Or.inl hx)).continuousWithinAt | theorem | Topology | [
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.GroupWithZero.Units.Equiv",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Homeomorph.Lemmas"
] | Mathlib/Topology/Algebra/GroupWithZero.lean | continuousOn_zpow₀ | null |
Filter.Tendsto.zpow₀ {f : α → G₀} {l : Filter α} {a : G₀} (hf : Tendsto f l (𝓝 a)) (m : ℤ)
(h : a ≠ 0 ∨ 0 ≤ m) : Tendsto (fun x => f x ^ m) l (𝓝 (a ^ m)) :=
(continuousAt_zpow₀ _ m h).tendsto.comp hf
variable {X : Type*} [TopologicalSpace X] {a : X} {s : Set X} {f : X → G₀}
@[fun_prop]
nonrec theorem ContinuousAt.zpow₀ (hf : ContinuousAt f a) (m : ℤ) (h : f a ≠ 0 ∨ 0 ≤ m) :
ContinuousAt (fun x => f x ^ m) a :=
hf.zpow₀ m h
nonrec theorem ContinuousWithinAt.zpow₀ (hf : ContinuousWithinAt f s a) (m : ℤ)
(h : f a ≠ 0 ∨ 0 ≤ m) : ContinuousWithinAt (fun x => f x ^ m) s a :=
hf.zpow₀ m h
@[fun_prop] | theorem | Topology | [
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.GroupWithZero.Units.Equiv",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Homeomorph.Lemmas"
] | Mathlib/Topology/Algebra/GroupWithZero.lean | Filter.Tendsto.zpow₀ | null |
ContinuousOn.zpow₀ (hf : ContinuousOn f s) (m : ℤ) (h : ∀ a ∈ s, f a ≠ 0 ∨ 0 ≤ m) :
ContinuousOn (fun x => f x ^ m) s := fun a ha => (hf a ha).zpow₀ m (h a ha)
@[continuity, fun_prop] | theorem | Topology | [
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.GroupWithZero.Units.Equiv",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Homeomorph.Lemmas"
] | Mathlib/Topology/Algebra/GroupWithZero.lean | ContinuousOn.zpow₀ | null |
Continuous.zpow₀ (hf : Continuous f) (m : ℤ) (h0 : ∀ a, f a ≠ 0 ∨ 0 ≤ m) :
Continuous fun x => f x ^ m :=
continuous_iff_continuousAt.2 fun x => (hf.tendsto x).zpow₀ m (h0 x) | theorem | Topology | [
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.GroupWithZero.Units.Equiv",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Homeomorph.Lemmas"
] | Mathlib/Topology/Algebra/GroupWithZero.lean | Continuous.zpow₀ | null |
@[to_additive]
continuous_mulIndicator (hs : ∀ a ∈ frontier s, f a = 1) (hf : ContinuousOn f (closure s)) :
Continuous (mulIndicator s f) := by
classical exact continuous_piecewise hs hf continuousOn_const
@[to_additive] | lemma | Topology | [
"Mathlib.Algebra.Notation.Indicator",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Clopen"
] | Mathlib/Topology/Algebra/Indicator.lean | continuous_mulIndicator | null |
protected Continuous.mulIndicator (hs : ∀ a ∈ frontier s, f a = 1) (hf : Continuous f) :
Continuous (mulIndicator s f) := by
classical exact hf.piecewise hs continuous_const
@[to_additive] | lemma | Topology | [
"Mathlib.Algebra.Notation.Indicator",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Clopen"
] | Mathlib/Topology/Algebra/Indicator.lean | Continuous.mulIndicator | null |
ContinuousOn.continuousAt_mulIndicator (hf : ContinuousOn f (interior s)) {x : α}
(hx : x ∉ frontier s) :
ContinuousAt (s.mulIndicator f) x := by
rw [← Set.mem_compl_iff, compl_frontier_eq_union_interior] at hx
obtain h | h := hx
· have hs : interior s ∈ 𝓝 x := mem_interior_iff_mem_nhds.mp (by rwa [interior_interior])
exact ContinuousAt.congr (hf.continuousAt hs) <| Filter.eventuallyEq_iff_exists_mem.mpr
⟨interior s, hs, Set.eqOn_mulIndicator.symm.mono interior_subset⟩
· exact ContinuousAt.congr continuousAt_const <| Filter.eventuallyEq_iff_exists_mem.mpr
⟨sᶜ, mem_interior_iff_mem_nhds.mp h, Set.eqOn_mulIndicator'.symm⟩
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Notation.Indicator",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Clopen"
] | Mathlib/Topology/Algebra/Indicator.lean | ContinuousOn.continuousAt_mulIndicator | null |
IsClopen.continuous_mulIndicator (hs : IsClopen s) (hf : Continuous f) :
Continuous (s.mulIndicator f) :=
hf.mulIndicator (by simp [isClopen_iff_frontier_eq_empty.mp hs]) | lemma | Topology | [
"Mathlib.Algebra.Notation.Indicator",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Clopen"
] | Mathlib/Topology/Algebra/Indicator.lean | IsClopen.continuous_mulIndicator | null |
IntermediateField.continuousSMul (M : IntermediateField K L) : ContinuousSMul M X :=
M.toSubfield.continuousSMul X | instance | Topology | [
"Mathlib.FieldTheory.IntermediateField.Adjoin.Defs",
"Mathlib.Topology.Algebra.Field"
] | Mathlib/Topology/Algebra/IntermediateField.lean | IntermediateField.continuousSMul | null |
IntermediateField.botContinuousSMul (M : IntermediateField K L) :
ContinuousSMul (⊥ : IntermediateField K L) M :=
Topology.IsInducing.continuousSMul (X := L) (N := (⊥ : IntermediateField K L)) (Y := M)
(M := (⊥ : IntermediateField K L)) Topology.IsInducing.subtypeVal continuous_id rfl | instance | Topology | [
"Mathlib.FieldTheory.IntermediateField.Adjoin.Defs",
"Mathlib.Topology.Algebra.Field"
] | Mathlib/Topology/Algebra/IntermediateField.lean | IntermediateField.botContinuousSMul | null |
@[mk_iff]
IsOpenUnits (M : Type*) [Monoid M] [TopologicalSpace M] : Prop where
isOpenEmbedding_unitsVal : IsOpenEmbedding (Units.val : Mˣ → M) | class | Topology | [
"Mathlib.RingTheory.Jacobson.Ideal",
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology"
] | Mathlib/Topology/Algebra/IsOpenUnits.lean | IsOpenUnits | We say that a topological monoid `M` has open units if `Mˣ` is open in `M` and
has the subspace topology (i.e. inverse is continuous).
Typical examples include monoids with discrete topology, topological groups (or fields),
and rings `R` equipped with the `I`-adic topology where `I ≤ J(R)`. |
IsOpenUnits.of_isAdic {R : Type*} [CommRing R] [TopologicalSpace R] [IsTopologicalRing R]
{I : Ideal R}
(hR : IsAdic I) (hI : I ≤ Ideal.jacobson ⊥) :
IsOpenUnits R := by
refine ⟨.of_continuous_injective_isOpenMap Units.continuous_val Units.val_injective ?_⟩
refine (IsTopologicalGroup.isOpenMap_iff_nhds_one (f := Units.coeHom R)).mpr ?_
rw [nhds_induced, nhds_prod_eq]
simp only [Units.embedProduct_apply, Units.val_one, inv_one, MulOpposite.op_one]
intro s hs
have H := hR ▸ Ideal.hasBasis_nhds_adic I 1
have := (H.prod (H.comap MulOpposite.opHomeomorph.symm))
simp only [Homeomorph.comap_nhds_eq, Homeomorph.symm_symm, MulOpposite.opHomeomorph_apply,
MulOpposite.op_one, and_self, Set.image_add_left] at this
have : ∃ n₁ n₂, ∀ (u : Rˣ), (-1 + u : R) ∈ I ^ n₁ → (-1 + u⁻¹ : R) ∈ I ^ n₂ → ↑u ∈ s := by
simpa [Set.subset_def, forall_comm (β := Rˣ), forall_comm (β := _ = _)] using
(((this.comap (Units.embedProduct R)).map (Units.coeHom R)).1 _).mp hs
obtain ⟨n, hn, hn'⟩ : ∃ n ≠ 0, ∀ (u : Rˣ), (-1 + u : R) ∈ I ^ n →
(-1 + u⁻¹ : R) ∈ I ^ n → ↑u ∈ s := by
obtain ⟨n₁, n₂, H⟩ := this
exact ⟨n₁ ⊔ n₂ ⊔ 1, by simp, fun u h₁ h₂ ↦ H u
(Ideal.pow_le_pow_right (by simp) h₁)
(Ideal.pow_le_pow_right (by simp) h₂)⟩
rw [H.1]
refine ⟨n, trivial, ?_⟩
rintro _ ⟨x, hx, rfl⟩
have := Ideal.mem_jacobson_bot.mp (hI (Ideal.pow_le_self hn hx)) 1
rw [mul_one, add_comm] at this
refine hn' this.unit (by simpa using hx) ?_
have : -1 + ↑this.unit⁻¹ = -this.unit⁻¹ * x := by
trans this.unit⁻¹ * (-(1 + x) + 1)
· rw [mul_add, mul_neg, IsUnit.val_inv_mul, mul_one]
· simp
rw [this]
exact Ideal.mul_mem_left _ _ hx | lemma | Topology | [
"Mathlib.RingTheory.Jacobson.Ideal",
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology"
] | Mathlib/Topology/Algebra/IsOpenUnits.lean | IsOpenUnits.of_isAdic | If `R` has the `I`-adic topology where `I` is contained in the Jacobson radical
(e.g. when `R` is complete or local), then `Rˣ` is an open subspace of `R`. |
_root_.IsLinearTopology where
hasBasis_submodule' : (𝓝 (0 : M)).HasBasis
(fun N : Submodule R M ↦ (N : Set M) ∈ 𝓝 0) (fun N : Submodule R M ↦ (N : Set M))
variable (R) in | class | Topology | [
"Mathlib.RingTheory.TwoSidedIdeal.Operations",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.OpenSubgroup"
] | Mathlib/Topology/Algebra/LinearTopology.lean | _root_.IsLinearTopology | Consider a (left-)module `M` over a ring `R`. A topology on `M` is *`R`-linear*
if the open sub-`R`-modules of `M` form a basis of neighborhoods of zero.
Typically one would also that the topology is invariant by translation (`ContinuousConstVAdd M M`),
or equivalently that `M` is a topological group, but we do not assume it for the definition.
In particular, we say that a topology on the ring `R` is *linear* if it is both
`R`-linear and `Rᵐᵒᵖ`-linear for the obvious module structures. To spell this in Lean,
simply use `[IsLinearTopology R R] [IsLinearTopology Rᵐᵒᵖ R]`. |
hasBasis_submodule [IsLinearTopology R M] : (𝓝 (0 : M)).HasBasis
(fun N : Submodule R M ↦ (N : Set M) ∈ 𝓝 0) (fun N : Submodule R M ↦ (N : Set M)) :=
IsLinearTopology.hasBasis_submodule'
variable (R) in | lemma | Topology | [
"Mathlib.RingTheory.TwoSidedIdeal.Operations",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.OpenSubgroup"
] | Mathlib/Topology/Algebra/LinearTopology.lean | hasBasis_submodule | null |
hasBasis_open_submodule [ContinuousAdd M] [IsLinearTopology R M] :
(𝓝 (0 : M)).HasBasis
(fun N : Submodule R M ↦ IsOpen (N : Set M)) (fun N : Submodule R M ↦ (N : Set M)) :=
hasBasis_submodule R |>.congr
(fun N ↦ ⟨N.toAddSubgroup.isOpen_of_mem_nhds, fun hN ↦ hN.mem_nhds (zero_mem N)⟩)
(fun _ _ ↦ rfl)
variable (R) in | lemma | Topology | [
"Mathlib.RingTheory.TwoSidedIdeal.Operations",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.OpenSubgroup"
] | Mathlib/Topology/Algebra/LinearTopology.lean | hasBasis_open_submodule | null |
mk_of_hasBasis' {ι : Sort*} {S : Type*} [SetLike S M]
[AddSubmonoidClass S M]
{p : ι → Prop} {s : ι → S}
(h : (𝓝 0).HasBasis p (fun i ↦ (s i : Set M)))
(hsmul : ∀ s : S, ∀ r : R, ∀ m ∈ s, r • m ∈ s) :
IsLinearTopology R M where
hasBasis_submodule' := h.to_hasBasis
(fun i hi ↦ ⟨
{ carrier := s i,
add_mem' := add_mem,
zero_mem' := zero_mem _,
smul_mem' := hsmul _},
h.mem_of_mem hi, subset_rfl⟩)
(fun _ ↦ h.mem_iff.mp)
variable (R) in | lemma | Topology | [
"Mathlib.RingTheory.TwoSidedIdeal.Operations",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.OpenSubgroup"
] | Mathlib/Topology/Algebra/LinearTopology.lean | mk_of_hasBasis' | A variant of `IsLinearTopology.mk_of_hasBasis` asking for an explicit proof that `S`
is a class of submodules instead of relying on (fragile) typeclass inference of `SMulCommClass`. |
mk_of_hasBasis {ι : Sort*} {S : Type*} [SetLike S M]
[SMulMemClass S R M] [AddSubmonoidClass S M]
{p : ι → Prop} {s : ι → S}
(h : (𝓝 0).HasBasis p (fun i ↦ (s i : Set M))) :
IsLinearTopology R M :=
mk_of_hasBasis' R h fun _ ↦ SMulMemClass.smul_mem | lemma | Topology | [
"Mathlib.RingTheory.TwoSidedIdeal.Operations",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.OpenSubgroup"
] | Mathlib/Topology/Algebra/LinearTopology.lean | mk_of_hasBasis | To show that `M` is linearly-topologized as an `R`-module, it suffices to show
that it has a basis of neighborhoods of zero made of `R`-submodules.
Note: for technical reasons detailed in the module docstring, Lean sometimes struggles to find the
right `SMulMemClass` instance. See `IsLinearTopology.mk_of_hasBasis'` for a more
explicit variant. |
_root_.isLinearTopology_iff_hasBasis_submodule :
IsLinearTopology R M ↔ (𝓝 0).HasBasis
(fun N : Submodule R M ↦ (N : Set M) ∈ 𝓝 0) (fun N : Submodule R M ↦ (N : Set M)) :=
⟨fun _ ↦ hasBasis_submodule R, fun h ↦ .mk_of_hasBasis R h⟩ | theorem | Topology | [
"Mathlib.RingTheory.TwoSidedIdeal.Operations",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.OpenSubgroup"
] | Mathlib/Topology/Algebra/LinearTopology.lean | _root_.isLinearTopology_iff_hasBasis_submodule | null |
_root_.isLinearTopology_iff_hasBasis_open_submodule [ContinuousAdd M] :
IsLinearTopology R M ↔ (𝓝 0).HasBasis
(fun N : Submodule R M ↦ IsOpen (N : Set M)) (fun N : Submodule R M ↦ (N : Set M)) :=
⟨fun _ ↦ hasBasis_open_submodule R, fun h ↦ .mk_of_hasBasis R h⟩ | theorem | Topology | [
"Mathlib.RingTheory.TwoSidedIdeal.Operations",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.OpenSubgroup"
] | Mathlib/Topology/Algebra/LinearTopology.lean | _root_.isLinearTopology_iff_hasBasis_open_submodule | null |
hasBasis_subbimodule [IsLinearTopology R M] [IsLinearTopology R' M] :
(𝓝 (0 : M)).HasBasis
(fun I : AddSubgroup M ↦ (I : Set M) ∈ 𝓝 0 ∧
(∀ r : R, ∀ x ∈ I, r • x ∈ I) ∧ (∀ r' : R', ∀ x ∈ I, r' • x ∈ I))
(fun I : AddSubgroup M ↦ (I : Set M)) := by
refine IsLinearTopology.hasBasis_submodule R |>.to_hasBasis (fun I hI ↦ ?_)
(fun I hI ↦ ⟨{I with smul_mem' := fun r x hx ↦ hI.2.1 r x hx}, hI.1, subset_rfl⟩)
rcases (hasBasis_submodule R').mem_iff.mp hI with ⟨J, hJ, J_sub_I⟩
set uR : Set R := univ -- Convenient to avoid type ascriptions
set uR' : Set R' := univ
have hRR : uR * uR ⊆ uR := subset_univ _
have hRI : uR • (I : Set M) ⊆ I := smul_subset_iff.mpr fun x _ i hi ↦ I.smul_mem x hi
have hR'J : uR' • (J : Set M) ⊆ J := smul_subset_iff.mpr fun x _ j hj ↦ J.smul_mem x hj
have hRJ : uR • (J : Set M) ⊆ I := subset_trans (smul_subset_smul_left J_sub_I) hRI
set S : Set M := J ∪ uR • J
have S_sub_I : S ⊆ I := union_subset J_sub_I hRJ
have hRS : uR • S ⊆ S := calc
uR • S = uR • (J : Set M) ∪ (uR * uR) • (J : Set M) := by simp_rw [S, smul_union, mul_smul]
_ ⊆ uR • (J : Set M) ∪ uR • (J : Set M) := by gcongr
_ = uR • (J : Set M) := union_self _
_ ⊆ S := subset_union_right
have hR'S : uR' • S ⊆ S := calc
uR' • S = uR' • (J : Set M) ∪ uR • uR' • (J : Set M) := by simp_rw [S, smul_union, smul_comm]
_ ⊆ J ∪ uR • J := by gcongr
_ = S := rfl
set A : AddSubgroup M := .closure S
have hRA : ∀ r : R, ∀ i ∈ A, r • i ∈ A := fun r i hi ↦ by
refine AddSubgroup.closure_induction (fun x hx => ?base) ?zero (fun x y _ _ hx hy ↦ ?add)
(fun x _ hx ↦ ?neg) hi
case base => exact AddSubgroup.subset_closure <| hRS <| Set.smul_mem_smul trivial hx
case zero => simp_rw [smul_zero]; exact zero_mem _
case add => simp_rw [smul_add]; exact add_mem hx hy
case neg => simp_rw [smul_neg]; exact neg_mem hx
have hR'A : ∀ r' : R', ∀ i ∈ A, r' • i ∈ A := fun r' i hi ↦ by
refine AddSubgroup.closure_induction (fun x hx => ?base) ?zero (fun x y _ _ hx hy ↦ ?add)
(fun x _ hx ↦ ?neg) hi
case base => exact AddSubgroup.subset_closure <| hR'S <| Set.smul_mem_smul trivial hx
case zero => simp_rw [smul_zero]; exact zero_mem _
case add => simp_rw [smul_add]; exact add_mem hx hy
case neg => simp_rw [smul_neg]; exact neg_mem hx
have A_sub_I : (A : Set M) ⊆ I := I.toAddSubgroup.closure_le.mpr S_sub_I
have J_sub_A : (J : Set M) ⊆ A := subset_trans subset_union_left AddSubgroup.subset_closure
exact ⟨A, ⟨mem_of_superset hJ J_sub_A, hRA, hR'A⟩, A_sub_I⟩
variable (R R') in
open Set Pointwise in | lemma | Topology | [
"Mathlib.RingTheory.TwoSidedIdeal.Operations",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.OpenSubgroup"
] | Mathlib/Topology/Algebra/LinearTopology.lean | hasBasis_subbimodule | The discrete topology on any `R`-module is `R`-linear. -/
instance [DiscreteTopology M] : IsLinearTopology R M :=
have : HasBasis (𝓝 0 : Filter M) (fun _ ↦ True) (fun (_ : Unit) ↦ (⊥ : Submodule R M)) := by
rw [nhds_discrete]
exact hasBasis_pure _
mk_of_hasBasis R this
variable (R R') in
open Set Pointwise in
/-- Assume that `M` is a module over two rings `R` and `R'`, and that its topology
is linear with respect to each of these rings. Then, it has a basis of neighborhoods of zero
made of sub-`(R, R')`-bimodules.
The proof is inspired by lemma 9 in [I. Kaplansky, *Topological Rings*](kaplansky_topological_1947).
TODO: Formalize the lemma in its full strength.
Note: due to the lack of a satisfying theory of sub-bimodules, we use `AddSubgroup`s with
extra conditions. |
hasBasis_open_subbimodule [ContinuousAdd M] [IsLinearTopology R M] [IsLinearTopology R' M] :
(𝓝 (0 : M)).HasBasis
(fun I : AddSubgroup M ↦ IsOpen (I : Set M) ∧
(∀ r : R, ∀ x ∈ I, r • x ∈ I) ∧ (∀ r' : R', ∀ x ∈ I, r' • x ∈ I))
(fun I : AddSubgroup M ↦ (I : Set M)) :=
hasBasis_subbimodule R R' |>.congr
(fun N ↦ and_congr_left' ⟨N.isOpen_of_mem_nhds, fun hN ↦ hN.mem_nhds (zero_mem N)⟩)
(fun _ _ ↦ rfl)
variable (R) in | lemma | Topology | [
"Mathlib.RingTheory.TwoSidedIdeal.Operations",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.OpenSubgroup"
] | Mathlib/Topology/Algebra/LinearTopology.lean | hasBasis_open_subbimodule | A variant of `IsLinearTopology.hasBasis_subbimodule` using `IsOpen I` instead of `I ∈ 𝓝 0`. |
tendsto_smul_zero [IsLinearTopology R M] {ι : Type*} {f : Filter ι}
(a : ι → R) (m : ι → M) (ha : Tendsto m f (𝓝 0)) :
Tendsto (a • m) f (𝓝 0) := by
rw [hasBasis_submodule R |>.tendsto_right_iff] at ha ⊢
intro I hI
filter_upwards [ha I hI] with i ai_mem
exact I.smul_mem _ ai_mem
variable (R) in | theorem | Topology | [
"Mathlib.RingTheory.TwoSidedIdeal.Operations",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.OpenSubgroup"
] | Mathlib/Topology/Algebra/LinearTopology.lean | tendsto_smul_zero | If `M` is a linearly topologized `R`-module and `i ↦ m i` tends to zero,
then `i ↦ a i • m i` still tends to zero for any family `a : ι → R`. |
_root_.IsCentralScalar.isLinearTopology_iff [Module Rᵐᵒᵖ M] [IsCentralScalar R M] :
IsLinearTopology Rᵐᵒᵖ M ↔ IsLinearTopology R M := by
constructor <;> intro H
· exact mk_of_hasBasis' R (IsLinearTopology.hasBasis_submodule Rᵐᵒᵖ)
fun S r m hm ↦ op_smul_eq_smul r m ▸ S.smul_mem _ hm
· exact mk_of_hasBasis' Rᵐᵒᵖ (IsLinearTopology.hasBasis_submodule R)
fun S r m hm ↦ unop_smul_eq_smul r m ▸ S.smul_mem _ hm | theorem | Topology | [
"Mathlib.RingTheory.TwoSidedIdeal.Operations",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.OpenSubgroup"
] | Mathlib/Topology/Algebra/LinearTopology.lean | _root_.IsCentralScalar.isLinearTopology_iff | If the left and right actions of `R` on `M` coincide, then a topology is `Rᵐᵒᵖ`-linear
if and only if it is `R`-linear. |
hasBasis_ideal [IsLinearTopology R R] :
(𝓝 0).HasBasis (fun I : Ideal R ↦ (I : Set R) ∈ 𝓝 0) (fun I : Ideal R ↦ (I : Set R)) :=
hasBasis_submodule R | theorem | Topology | [
"Mathlib.RingTheory.TwoSidedIdeal.Operations",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.OpenSubgroup"
] | Mathlib/Topology/Algebra/LinearTopology.lean | hasBasis_ideal | null |
hasBasis_open_ideal [ContinuousAdd R] [IsLinearTopology R R] :
(𝓝 0).HasBasis (fun I : Ideal R ↦ IsOpen (I : Set R)) (fun I : Ideal R ↦ (I : Set R)) :=
hasBasis_open_submodule R | theorem | Topology | [
"Mathlib.RingTheory.TwoSidedIdeal.Operations",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.OpenSubgroup"
] | Mathlib/Topology/Algebra/LinearTopology.lean | hasBasis_open_ideal | null |
_root_.isLinearTopology_iff_hasBasis_ideal :
IsLinearTopology R R ↔ (𝓝 0).HasBasis
(fun I : Ideal R ↦ (I : Set R) ∈ 𝓝 0) (fun I : Ideal R ↦ (I : Set R)) :=
isLinearTopology_iff_hasBasis_submodule | theorem | Topology | [
"Mathlib.RingTheory.TwoSidedIdeal.Operations",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.OpenSubgroup"
] | Mathlib/Topology/Algebra/LinearTopology.lean | _root_.isLinearTopology_iff_hasBasis_ideal | null |
_root_.isLinearTopology_iff_hasBasis_open_ideal [IsTopologicalRing R] :
IsLinearTopology R R ↔ (𝓝 0).HasBasis
(fun I : Ideal R ↦ IsOpen (I : Set R)) (fun I : Ideal R ↦ (I : Set R)) :=
isLinearTopology_iff_hasBasis_open_submodule | theorem | Topology | [
"Mathlib.RingTheory.TwoSidedIdeal.Operations",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.OpenSubgroup"
] | Mathlib/Topology/Algebra/LinearTopology.lean | _root_.isLinearTopology_iff_hasBasis_open_ideal | null |
hasBasis_right_ideal [IsLinearTopology Rᵐᵒᵖ R] :
(𝓝 0).HasBasis (fun I : Submodule Rᵐᵒᵖ R ↦ (I : Set R) ∈ 𝓝 0) (fun I ↦ (I : Set R)) :=
hasBasis_submodule Rᵐᵒᵖ
open Set Pointwise in | theorem | Topology | [
"Mathlib.RingTheory.TwoSidedIdeal.Operations",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.OpenSubgroup"
] | Mathlib/Topology/Algebra/LinearTopology.lean | hasBasis_right_ideal | null |
hasBasis_twoSidedIdeal [IsLinearTopology R R] [IsLinearTopology Rᵐᵒᵖ R] :
(𝓝 (0 : R)).HasBasis (fun I : TwoSidedIdeal R ↦ (I : Set R) ∈ 𝓝 0)
(fun I : TwoSidedIdeal R ↦ (I : Set R)) :=
hasBasis_subbimodule R Rᵐᵒᵖ |>.to_hasBasis
(fun I ⟨hI, hRI, hRI'⟩ ↦ ⟨.mk' I (zero_mem _) add_mem neg_mem (hRI _ _) (hRI' _ _),
by simpa using hI, by simp⟩)
(fun I hI ↦ ⟨I.asIdeal.toAddSubgroup,
⟨hI, I.mul_mem_left, fun r x hx ↦ I.mul_mem_right x (r.unop) hx⟩, subset_rfl⟩) | lemma | Topology | [
"Mathlib.RingTheory.TwoSidedIdeal.Operations",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.OpenSubgroup"
] | Mathlib/Topology/Algebra/LinearTopology.lean | hasBasis_twoSidedIdeal | If a ring `R` is linearly ordered as a left *and* right module over itself,
then it has a basis of neighborhoods of zero made of *two-sided* ideals.
This is usually called a *linearly topologized ring*, but we do not add a specific spelling:
you should use `[IsLinearTopology R R] [IsLinearTopology Rᵐᵒᵖ R]` instead. |
hasBasis_open_twoSidedIdeal [ContinuousAdd R]
[IsLinearTopology R R] [IsLinearTopology Rᵐᵒᵖ R] :
(𝓝 (0 : R)).HasBasis
(fun I : TwoSidedIdeal R ↦ IsOpen (I : Set R)) (fun I : TwoSidedIdeal R ↦ (I : Set R)) :=
hasBasis_twoSidedIdeal.congr
(fun I ↦ ⟨I.asIdeal.toAddSubgroup.isOpen_of_mem_nhds, fun hI ↦ hI.mem_nhds (zero_mem I)⟩)
(fun _ _ ↦ rfl) | lemma | Topology | [
"Mathlib.RingTheory.TwoSidedIdeal.Operations",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.OpenSubgroup"
] | Mathlib/Topology/Algebra/LinearTopology.lean | hasBasis_open_twoSidedIdeal | null |
_root_.isLinearTopology_iff_hasBasis_twoSidedIdeal :
IsLinearTopology R R ∧ IsLinearTopology Rᵐᵒᵖ R ↔
(𝓝 0).HasBasis
(fun I : TwoSidedIdeal R ↦ (I : Set R) ∈ 𝓝 0) (fun I : TwoSidedIdeal R ↦ (I : Set R)) :=
⟨fun ⟨_, _⟩ ↦ hasBasis_twoSidedIdeal, fun h ↦
⟨.mk_of_hasBasis' R h fun I r x hx ↦ I.mul_mem_left r x hx,
.mk_of_hasBasis' Rᵐᵒᵖ h fun I r x hx ↦ I.mul_mem_right x r.unop hx⟩⟩ | theorem | Topology | [
"Mathlib.RingTheory.TwoSidedIdeal.Operations",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.OpenSubgroup"
] | Mathlib/Topology/Algebra/LinearTopology.lean | _root_.isLinearTopology_iff_hasBasis_twoSidedIdeal | null |
_root_.isLinearTopology_iff_hasBasis_open_twoSidedIdeal [ContinuousAdd R] :
IsLinearTopology R R ∧ IsLinearTopology Rᵐᵒᵖ R ↔ (𝓝 0).HasBasis
(fun I : TwoSidedIdeal R ↦ IsOpen (I : Set R)) (fun I : TwoSidedIdeal R ↦ (I : Set R)) :=
⟨fun ⟨_, _⟩ ↦ hasBasis_open_twoSidedIdeal, fun h ↦
⟨.mk_of_hasBasis' R h fun I r x hx ↦ I.mul_mem_left r x hx,
.mk_of_hasBasis' Rᵐᵒᵖ h fun I r x hx ↦ I.mul_mem_right x r.unop hx⟩⟩ | theorem | Topology | [
"Mathlib.RingTheory.TwoSidedIdeal.Operations",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.OpenSubgroup"
] | Mathlib/Topology/Algebra/LinearTopology.lean | _root_.isLinearTopology_iff_hasBasis_open_twoSidedIdeal | null |
tendsto_mul_zero_of_left [IsLinearTopology Rᵐᵒᵖ R] {ι : Type*} {f : Filter ι}
(a b : ι → R) (ha : Tendsto a f (𝓝 0)) :
Tendsto (a * b) f (𝓝 0) :=
tendsto_smul_zero (R := Rᵐᵒᵖ) _ _ ha | theorem | Topology | [
"Mathlib.RingTheory.TwoSidedIdeal.Operations",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.OpenSubgroup"
] | Mathlib/Topology/Algebra/LinearTopology.lean | tendsto_mul_zero_of_left | null |
tendsto_mul_zero_of_right [IsLinearTopology R R] {ι : Type*} {f : Filter ι}
(a b : ι → R) (hb : Tendsto b f (𝓝 0)) :
Tendsto (a * b) f (𝓝 0) :=
tendsto_smul_zero (R := R) _ _ hb | theorem | Topology | [
"Mathlib.RingTheory.TwoSidedIdeal.Operations",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.OpenSubgroup"
] | Mathlib/Topology/Algebra/LinearTopology.lean | tendsto_mul_zero_of_right | null |
Localization.ringTopology : RingTopology (Localization M) :=
RingTopology.coinduced (Localization.monoidOf M).toFun | def | Topology | [
"Mathlib.GroupTheory.MonoidLocalization.Basic",
"Mathlib.RingTheory.OreLocalization.Ring",
"Mathlib.Topology.Algebra.Ring.Basic"
] | Mathlib/Topology/Algebra/Localization.lean | Localization.ringTopology | The ring topology on `Localization M` coinduced from the natural homomorphism sending `x : R`
to the equivalence class of `(x, 1)`. |
@[to_additive (attr := continuity, fun_prop)]
continuous_one [TopologicalSpace M] [One M] : Continuous (1 : X → M) :=
@continuous_const _ _ _ _ 1 | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Algebra.Group.Submonoid.Basic",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Monoid.lean | continuous_one | null |
@[to_additive]
ContinuousMul.to_continuousSMul : ContinuousSMul M M :=
⟨continuous_mul⟩
@[to_additive] | instance | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Algebra.Group.Submonoid.Basic",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Monoid.lean | ContinuousMul.to_continuousSMul | null |
ContinuousMul.to_continuousSMul_op : ContinuousSMul Mᵐᵒᵖ M :=
⟨show Continuous ((fun p : M × M => p.1 * p.2) ∘ Prod.swap ∘ Prod.map MulOpposite.unop id) from
continuous_mul.comp <|
continuous_swap.comp <| Continuous.prodMap MulOpposite.continuous_unop continuous_id⟩
@[to_additive] | instance | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Algebra.Group.Submonoid.Basic",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Monoid.lean | ContinuousMul.to_continuousSMul_op | null |
ContinuousMul.induced {α : Type*} {β : Type*} {F : Type*} [FunLike F α β] [Mul α]
[Mul β] [MulHomClass F α β] [tβ : TopologicalSpace β] [ContinuousMul β] (f : F) :
@ContinuousMul α (tβ.induced f) _ := by
let tα := tβ.induced f
refine ⟨continuous_induced_rng.2 ?_⟩
simp only [Function.comp_def, map_mul]
fun_prop
@[to_additive (attr := continuity)] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Algebra.Group.Submonoid.Basic",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Monoid.lean | ContinuousMul.induced | null |
continuous_mul_left (a : M) : Continuous fun b : M => a * b :=
continuous_const.mul continuous_id
@[to_additive (attr := continuity)] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Algebra.Group.Submonoid.Basic",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Monoid.lean | continuous_mul_left | null |
continuous_mul_right (a : M) : Continuous fun b : M => b * a :=
continuous_id.mul continuous_const
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Algebra.Group.Submonoid.Basic",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Monoid.lean | continuous_mul_right | null |
tendsto_mul {a b : M} : Tendsto (fun p : M × M => p.fst * p.snd) (𝓝 (a, b)) (𝓝 (a * b)) :=
continuous_iff_continuousAt.mp ContinuousMul.continuous_mul (a, b)
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Algebra.Group.Submonoid.Basic",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Monoid.lean | tendsto_mul | null |
Filter.Tendsto.const_mul (b : M) {c : M} {f : α → M} {l : Filter α}
(h : Tendsto (fun k : α => f k) l (𝓝 c)) : Tendsto (fun k : α => b * f k) l (𝓝 (b * c)) :=
tendsto_const_nhds.mul h
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Algebra.Group.Submonoid.Basic",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Monoid.lean | Filter.Tendsto.const_mul | null |
Filter.Tendsto.mul_const (b : M) {c : M} {f : α → M} {l : Filter α}
(h : Tendsto (fun k : α => f k) l (𝓝 c)) : Tendsto (fun k : α => f k * b) l (𝓝 (c * b)) :=
h.mul tendsto_const_nhds
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Algebra.Group.Submonoid.Basic",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Monoid.lean | Filter.Tendsto.mul_const | null |
le_nhds_mul (a b : M) : 𝓝 a * 𝓝 b ≤ 𝓝 (a * b) := by
rw [← map₂_mul, ← map_uncurry_prod, ← nhds_prod_eq]
exact continuous_mul.tendsto _
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Algebra.Group.Submonoid.Basic",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Monoid.lean | le_nhds_mul | null |
nhds_one_mul_nhds {M} [MulOneClass M] [TopologicalSpace M] [ContinuousMul M] (a : M) :
𝓝 (1 : M) * 𝓝 a = 𝓝 a :=
((le_nhds_mul _ _).trans_eq <| congr_arg _ (one_mul a)).antisymm <|
le_mul_of_one_le_left' <| pure_le_nhds 1
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Algebra.Group.Submonoid.Basic",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Monoid.lean | nhds_one_mul_nhds | null |
nhds_mul_nhds_one {M} [MulOneClass M] [TopologicalSpace M] [ContinuousMul M] (a : M) :
𝓝 a * 𝓝 1 = 𝓝 a :=
((le_nhds_mul _ _).trans_eq <| congr_arg _ (mul_one a)).antisymm <|
le_mul_of_one_le_right' <| pure_le_nhds 1 | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Algebra.Group.Submonoid.Basic",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Monoid.lean | nhds_mul_nhds_one | null |
Filter.TendstoNhdsWithinIoi.const_mul [PosMulStrictMono 𝕜] (h : Tendsto f l (𝓝[>] c)) :
Tendsto (fun a => b * f a) l (𝓝[>] (b * c)) :=
tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within _
((tendsto_nhds_of_tendsto_nhdsWithin h).const_mul b) <|
(tendsto_nhdsWithin_iff.mp h).2.mono fun _ _ => by rw [Set.mem_Ioi] at *; gcongr; assumption | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Algebra.Group.Submonoid.Basic",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Monoid.lean | Filter.TendstoNhdsWithinIoi.const_mul | null |
Filter.TendstoNhdsWithinIio.const_mul [PosMulStrictMono 𝕜] (h : Tendsto f l (𝓝[<] c)) :
Tendsto (fun a => b * f a) l (𝓝[<] (b * c)) :=
tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within _
((tendsto_nhds_of_tendsto_nhdsWithin h).const_mul b) <|
(tendsto_nhdsWithin_iff.mp h).2.mono fun _ _ => by rw [Set.mem_Iio] at *; gcongr; assumption | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Algebra.Group.Submonoid.Basic",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Monoid.lean | Filter.TendstoNhdsWithinIio.const_mul | null |
Filter.TendstoNhdsWithinIoi.mul_const [MulPosStrictMono 𝕜] (h : Tendsto f l (𝓝[>] c)) :
Tendsto (fun a => f a * b) l (𝓝[>] (c * b)) :=
tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within _
((tendsto_nhds_of_tendsto_nhdsWithin h).mul_const b) <|
(tendsto_nhdsWithin_iff.mp h).2.mono fun _ _ => by rw [Set.mem_Ioi] at *; gcongr; assumption | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Algebra.Group.Submonoid.Basic",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Monoid.lean | Filter.TendstoNhdsWithinIoi.mul_const | null |
Filter.TendstoNhdsWithinIio.mul_const [MulPosStrictMono 𝕜] (h : Tendsto f l (𝓝[<] c)) :
Tendsto (fun a => f a * b) l (𝓝[<] (c * b)) :=
tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within _
((tendsto_nhds_of_tendsto_nhdsWithin h).mul_const b) <|
(tendsto_nhdsWithin_iff.mp h).2.mono fun _ _ => by rw [Set.mem_Iio] at *; gcongr; assumption | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Algebra.Group.Submonoid.Basic",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Monoid.lean | Filter.TendstoNhdsWithinIio.mul_const | null |
@[to_additive]
protected Specializes.mul {a b c d : M} (hab : a ⤳ b) (hcd : c ⤳ d) : (a * c) ⤳ (b * d) :=
hab.smul hcd
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Algebra.Group.Submonoid.Basic",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Monoid.lean | Specializes.mul | null |
protected Inseparable.mul {a b c d : M} (hab : Inseparable a b) (hcd : Inseparable c d) :
Inseparable (a * c) (b * d) :=
hab.smul hcd
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Algebra.Group.Submonoid.Basic",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Monoid.lean | Inseparable.mul | null |
protected Specializes.pow {M : Type*} [Monoid M] [TopologicalSpace M] [ContinuousMul M]
{a b : M} (h : a ⤳ b) (n : ℕ) : (a ^ n) ⤳ (b ^ n) :=
Nat.recOn n (by simp only [pow_zero, specializes_rfl]) fun _ ihn ↦ by
simpa only [pow_succ] using ihn.mul h
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Algebra.Group.Submonoid.Basic",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Monoid.lean | Specializes.pow | null |
protected Inseparable.pow {M : Type*} [Monoid M] [TopologicalSpace M] [ContinuousMul M]
{a b : M} (h : Inseparable a b) (n : ℕ) : Inseparable (a ^ n) (b ^ n) :=
(h.specializes.pow n).antisymm (h.specializes'.pow n) | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Algebra.Group.Submonoid.Basic",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Monoid.lean | Inseparable.pow | null |
@[to_additive (attr := simps)
/-- Construct an additive unit from limits of additive units and their negatives. -/]
Filter.Tendsto.units [TopologicalSpace N] [Monoid N] [ContinuousMul N] [T2Space N]
{f : ι → Nˣ} {r₁ r₂ : N} {l : Filter ι} [l.NeBot] (h₁ : Tendsto (fun x => ↑(f x)) l (𝓝 r₁))
(h₂ : Tendsto (fun x => ↑(f x)⁻¹) l (𝓝 r₂)) : Nˣ where
val := r₁
inv := r₂
val_inv := by
symm
simpa using h₁.mul h₂
inv_val := by
symm
simpa using h₂.mul h₁
@[to_additive] | def | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Algebra.Group.Submonoid.Basic",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Monoid.lean | Filter.Tendsto.units | Construct a unit from limits of units and their inverses. |
Prod.continuousMul [TopologicalSpace N] [Mul N] [ContinuousMul N] :
ContinuousMul (M × N) :=
⟨by apply Continuous.prodMk <;> fun_prop⟩
@[to_additive] | instance | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Algebra.Group.Submonoid.Basic",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Monoid.lean | Prod.continuousMul | null |
Pi.continuousMul {C : ι → Type*} [∀ i, TopologicalSpace (C i)] [∀ i, Mul (C i)]
[∀ i, ContinuousMul (C i)] : ContinuousMul (∀ i, C i) where
continuous_mul :=
continuous_pi fun i => (continuous_apply i).fst'.mul (continuous_apply i).snd' | instance | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Algebra.Group.Submonoid.Basic",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Monoid.lean | Pi.continuousMul | null |
@[to_additive /-- A version of `Pi.continuousAdd` for non-dependent functions. It is needed
because sometimes Lean fails to use `Pi.continuousAdd` for non-dependent functions. -/]
Pi.continuousMul' : ContinuousMul (ι → M) :=
Pi.continuousMul
@[to_additive] | instance | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Algebra.Group.Submonoid.Basic",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Monoid.lean | Pi.continuousMul' | A version of `Pi.continuousMul` for non-dependent functions. It is needed because sometimes
Lean 3 fails to use `Pi.continuousMul` for non-dependent functions. |
@[to_additive]
ContinuousMul.of_nhds_one {M : Type u} [Monoid M] [TopologicalSpace M]
(hmul : Tendsto (uncurry ((· * ·) : M → M → M)) (𝓝 1 ×ˢ 𝓝 1) <| 𝓝 1)
(hleft : ∀ x₀ : M, 𝓝 x₀ = map (fun x => x₀ * x) (𝓝 1))
(hright : ∀ x₀ : M, 𝓝 x₀ = map (fun x => x * x₀) (𝓝 1)) : ContinuousMul M :=
⟨by
rw [continuous_iff_continuousAt]
rintro ⟨x₀, y₀⟩
have key : (fun p : M × M => x₀ * p.1 * (p.2 * y₀)) =
((fun x => x₀ * x) ∘ fun x => x * y₀) ∘ uncurry (· * ·) := by
ext p
simp [uncurry, mul_assoc]
have key₂ : ((fun x => x₀ * x) ∘ fun x => y₀ * x) = fun x => x₀ * y₀ * x := by
ext x
simp [mul_assoc]
calc
map (uncurry (· * ·)) (𝓝 (x₀, y₀)) = map (uncurry (· * ·)) (𝓝 x₀ ×ˢ 𝓝 y₀) := by
rw [nhds_prod_eq]
_ = map (fun p : M × M => x₀ * p.1 * (p.2 * y₀)) (𝓝 1 ×ˢ 𝓝 1) := by
unfold uncurry
rw [hleft x₀, hright y₀, prod_map_map_eq, Filter.map_map, Function.comp_def]
_ = map ((fun x => x₀ * x) ∘ fun x => x * y₀) (map (uncurry (· * ·)) (𝓝 1 ×ˢ 𝓝 1)) := by
rw [key, ← Filter.map_map]
_ ≤ map ((fun x : M => x₀ * x) ∘ fun x => x * y₀) (𝓝 1) := map_mono hmul
_ = 𝓝 (x₀ * y₀) := by
rw [← Filter.map_map, ← hright, hleft y₀, Filter.map_map, key₂, ← hleft]⟩
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Algebra.Group.Submonoid.Basic",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Monoid.lean | ContinuousMul.of_nhds_one | null |
continuousMul_of_comm_of_nhds_one (M : Type u) [CommMonoid M] [TopologicalSpace M]
(hmul : Tendsto (uncurry ((· * ·) : M → M → M)) (𝓝 1 ×ˢ 𝓝 1) (𝓝 1))
(hleft : ∀ x₀ : M, 𝓝 x₀ = map (fun x => x₀ * x) (𝓝 1)) : ContinuousMul M := by
apply ContinuousMul.of_nhds_one hmul hleft
intro x₀
simp_rw [mul_comm, hleft x₀] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Algebra.Group.Submonoid.Basic",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Monoid.lean | continuousMul_of_comm_of_nhds_one | null |
@[to_additive]
isClosed_setOf_map_one [One M₁] [One M₂] : IsClosed { f : M₁ → M₂ | f 1 = 1 } :=
isClosed_eq (continuous_apply 1) continuous_const
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Algebra.Group.Submonoid.Basic",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Monoid.lean | isClosed_setOf_map_one | null |
isClosed_setOf_map_mul [Mul M₁] [Mul M₂] [ContinuousMul M₂] :
IsClosed { f : M₁ → M₂ | ∀ x y, f (x * y) = f x * f y } := by
simp only [setOf_forall]
exact isClosed_iInter fun x ↦ isClosed_iInter fun y ↦
isClosed_eq (continuous_apply _) (by fun_prop) | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Algebra.Group.Submonoid.Basic",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Monoid.lean | isClosed_setOf_map_mul | null |
@[to_additive (attr := simps -fullyApplied)
/-- Construct a bundled additive semigroup homomorphism `M₁ →ₙ+ M₂` from a function `f`
and a proof that it belongs to the closure of the range of the coercion from `M₁ →ₙ+ M₂` (or another
type of bundled homomorphisms that has an `AddHomClass` instance) to `M₁ → M₂`. -/]
mulHomOfMemClosureRangeCoe (f : M₁ → M₂)
(hf : f ∈ closure (range fun (f : F) (x : M₁) => f x)) : M₁ →ₙ* M₂ where
toFun := f
map_mul' := (isClosed_setOf_map_mul M₁ M₂).closure_subset_iff.2 (range_subset_iff.2 map_mul) hf | def | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Algebra.Group.Submonoid.Basic",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Monoid.lean | mulHomOfMemClosureRangeCoe | Construct a bundled semigroup homomorphism `M₁ →ₙ* M₂` from a function `f` and a proof that it
belongs to the closure of the range of the coercion from `M₁ →ₙ* M₂` (or another type of bundled
homomorphisms that has a `MulHomClass` instance) to `M₁ → M₂`. |
@[to_additive (attr := simps! -fullyApplied)
/-- Construct a bundled additive semigroup homomorphism from a pointwise limit of additive
semigroup homomorphisms -/]
mulHomOfTendsto (f : M₁ → M₂) (g : α → F) [l.NeBot]
(h : Tendsto (fun a x => g a x) l (𝓝 f)) : M₁ →ₙ* M₂ :=
mulHomOfMemClosureRangeCoe f <|
mem_closure_of_tendsto h <| Eventually.of_forall fun _ => mem_range_self _
variable (M₁ M₂)
@[to_additive] | def | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Algebra.Group.Submonoid.Basic",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Monoid.lean | mulHomOfTendsto | Construct a bundled semigroup homomorphism from a pointwise limit of semigroup homomorphisms. |
MulHom.isClosed_range_coe : IsClosed (Set.range ((↑) : (M₁ →ₙ* M₂) → M₁ → M₂)) :=
isClosed_of_closure_subset fun f hf => ⟨mulHomOfMemClosureRangeCoe f hf, rfl⟩ | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Algebra.Group.Submonoid.Basic",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Monoid.lean | MulHom.isClosed_range_coe | null |
@[to_additive (attr := simps -fullyApplied)
/-- Construct a bundled additive monoid homomorphism `M₁ →+ M₂` from a function `f`
and a proof that it belongs to the closure of the range of the coercion from `M₁ →+ M₂` (or another
type of bundled homomorphisms that has an `AddMonoidHomClass` instance) to `M₁ → M₂`. -/]
monoidHomOfMemClosureRangeCoe (f : M₁ → M₂)
(hf : f ∈ closure (range fun (f : F) (x : M₁) => f x)) : M₁ →* M₂ where
toFun := f
map_one' := (isClosed_setOf_map_one M₁ M₂).closure_subset_iff.2 (range_subset_iff.2 map_one) hf
map_mul' := (isClosed_setOf_map_mul M₁ M₂).closure_subset_iff.2 (range_subset_iff.2 map_mul) hf | def | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Algebra.Group.Submonoid.Basic",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Monoid.lean | monoidHomOfMemClosureRangeCoe | Construct a bundled monoid homomorphism `M₁ →* M₂` from a function `f` and a proof that it
belongs to the closure of the range of the coercion from `M₁ →* M₂` (or another type of bundled
homomorphisms that has a `MonoidHomClass` instance) to `M₁ → M₂`. |
@[to_additive (attr := simps! -fullyApplied)
/-- Construct a bundled additive monoid homomorphism from a pointwise limit of additive
monoid homomorphisms -/]
monoidHomOfTendsto (f : M₁ → M₂) (g : α → F) [l.NeBot]
(h : Tendsto (fun a x => g a x) l (𝓝 f)) : M₁ →* M₂ :=
monoidHomOfMemClosureRangeCoe f <|
mem_closure_of_tendsto h <| Eventually.of_forall fun _ => mem_range_self _
variable (M₁ M₂)
@[to_additive] | def | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Algebra.Group.Submonoid.Basic",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Monoid.lean | monoidHomOfTendsto | Construct a bundled monoid homomorphism from a pointwise limit of monoid homomorphisms. |
MonoidHom.isClosed_range_coe : IsClosed (Set.range ((↑) : (M₁ →* M₂) → M₁ → M₂)) :=
isClosed_of_closure_subset fun f hf => ⟨monoidHomOfMemClosureRangeCoe f hf, rfl⟩ | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Algebra.Group.Submonoid.Basic",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Monoid.lean | MonoidHom.isClosed_range_coe | null |
@[to_additive]
Topology.IsInducing.continuousMul {M N F : Type*} [Mul M] [Mul N] [FunLike F M N]
[MulHomClass F M N] [TopologicalSpace M] [TopologicalSpace N] [ContinuousMul N] (f : F)
(hf : IsInducing f) : ContinuousMul M :=
⟨(hf.continuousSMul hf.continuous (map_mul f _ _)).1⟩
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Algebra.Group.Submonoid.Basic",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Monoid.lean | Topology.IsInducing.continuousMul | null |
continuousMul_induced {M N F : Type*} [Mul M] [Mul N] [FunLike F M N] [MulHomClass F M N]
[TopologicalSpace N] [ContinuousMul N] (f : F) : @ContinuousMul M (induced f ‹_›) _ :=
letI := induced f ‹_›
IsInducing.continuousMul f ⟨rfl⟩
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Algebra.Group.Submonoid.Basic",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Monoid.lean | continuousMul_induced | null |
Subsemigroup.continuousMul [TopologicalSpace M] [Semigroup M] [ContinuousMul M]
(S : Subsemigroup M) : ContinuousMul S :=
IsInducing.continuousMul ({ toFun := (↑), map_mul' := fun _ _ => rfl} : MulHom S M) ⟨rfl⟩
@[to_additive] | instance | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Algebra.Group.Submonoid.Basic",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Monoid.lean | Subsemigroup.continuousMul | null |
Submonoid.continuousMul [TopologicalSpace M] [Monoid M] [ContinuousMul M]
(S : Submonoid M) : ContinuousMul S :=
S.toSubsemigroup.continuousMul | instance | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Algebra.Group.Submonoid.Basic",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Monoid.lean | Submonoid.continuousMul | null |
exists_mem_nhds_zero_mul_subset
{K U : Set M} (hK : IsCompact K) (hU : U ∈ 𝓝 0) : ∃ V ∈ 𝓝 0, K * V ⊆ U := by
refine hK.induction_on ?_ ?_ ?_ ?_
· exact ⟨univ, by simp⟩
· rintro s t hst ⟨V, hV, hV'⟩
exact ⟨V, hV, (mul_subset_mul_right hst).trans hV'⟩
· rintro s t ⟨V, V_in, hV'⟩ ⟨W, W_in, hW'⟩
use V ∩ W, inter_mem V_in W_in
rw [union_mul]
exact
union_subset ((mul_subset_mul_left V.inter_subset_left).trans hV')
((mul_subset_mul_left V.inter_subset_right).trans hW')
· intro x hx
have := tendsto_mul (show U ∈ 𝓝 (x * 0) by simpa using hU)
rw [nhds_prod_eq, mem_map, mem_prod_iff] at this
rcases this with ⟨t, ht, s, hs, h⟩
rw [← image_subset_iff, image_mul_prod] at h
exact ⟨t, mem_nhdsWithin_of_mem_nhds ht, s, hs, h⟩ | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Algebra.Group.Submonoid.Basic",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Monoid.lean | exists_mem_nhds_zero_mul_subset | null |
tendsto_mul_nhds_zero_prod_of_disjoint_cocompact {l : Filter M}
(hl : Disjoint l (cocompact M)) :
Tendsto (fun x : M × M ↦ x.1 * x.2) (𝓝 0 ×ˢ l) (𝓝 0) := calc
map (fun x : M × M ↦ x.1 * x.2) (𝓝 0 ×ˢ l)
_ ≤ map (fun x : M × M ↦ x.1 * x.2) (𝓝ˢ ({0} ×ˢ Set.univ)) :=
map_mono <| nhds_prod_le_of_disjoint_cocompact 0 hl
_ ≤ 𝓝 0 := continuous_mul.tendsto_nhdsSet_nhds fun _ ⟨hx, _⟩ ↦ mul_eq_zero_of_left hx _ | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Algebra.Group.Submonoid.Basic",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Monoid.lean | tendsto_mul_nhds_zero_prod_of_disjoint_cocompact | Let `M` be a topological space with a continuous multiplication operation and a `0`.
Let `l` be a filter on `M` which is disjoint from the cocompact filter. Then, the multiplication map
`M × M → M` tends to zero on the filter product `𝓝 0 ×ˢ l`. |
tendsto_mul_prod_nhds_zero_of_disjoint_cocompact {l : Filter M}
(hl : Disjoint l (cocompact M)) :
Tendsto (fun x : M × M ↦ x.1 * x.2) (l ×ˢ 𝓝 0) (𝓝 0) := calc
map (fun x : M × M ↦ x.1 * x.2) (l ×ˢ 𝓝 0)
_ ≤ map (fun x : M × M ↦ x.1 * x.2) (𝓝ˢ (Set.univ ×ˢ {0})) :=
map_mono <| prod_nhds_le_of_disjoint_cocompact 0 hl
_ ≤ 𝓝 0 := continuous_mul.tendsto_nhdsSet_nhds fun _ ⟨_, hx⟩ ↦ mul_eq_zero_of_right _ hx | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Algebra.Group.Submonoid.Basic",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Monoid.lean | tendsto_mul_prod_nhds_zero_of_disjoint_cocompact | Let `M` be a topological space with a continuous multiplication operation and a `0`.
Let `l` be a filter on `M` which is disjoint from the cocompact filter. Then, the multiplication map
`M × M → M` tends to zero on the filter product `l ×ˢ 𝓝 0`. |
tendsto_mul_coprod_nhds_zero_inf_of_disjoint_cocompact {l : Filter (M × M)}
(hl : Disjoint l (cocompact (M × M))) :
Tendsto (fun x : M × M ↦ x.1 * x.2) ((𝓝 0).coprod (𝓝 0) ⊓ l) (𝓝 0) := by
have := calc
(𝓝 0).coprod (𝓝 0) ⊓ l
_ ≤ (𝓝 0).coprod (𝓝 0) ⊓ map Prod.fst l ×ˢ map Prod.snd l :=
inf_le_inf_left _ le_prod_map_fst_snd
_ ≤ 𝓝 0 ×ˢ map Prod.snd l ⊔ map Prod.fst l ×ˢ 𝓝 0 :=
coprod_inf_prod_le _ _ _ _
apply (Tendsto.sup _ _).mono_left this
· apply tendsto_mul_nhds_zero_prod_of_disjoint_cocompact
exact disjoint_map_cocompact continuous_snd hl
· apply tendsto_mul_prod_nhds_zero_of_disjoint_cocompact
exact disjoint_map_cocompact continuous_fst hl | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Algebra.Group.Submonoid.Basic",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Monoid.lean | tendsto_mul_coprod_nhds_zero_inf_of_disjoint_cocompact | Let `M` be a topological space with a continuous multiplication operation and a `0`.
Let `l` be a filter on `M × M` which is disjoint from the cocompact filter. Then, the multiplication
map `M × M → M` tends to zero on `(𝓝 0).coprod (𝓝 0) ⊓ l`. |
tendsto_mul_nhds_zero_of_disjoint_cocompact {l : Filter (M × M)}
(hl : Disjoint l (cocompact (M × M))) (h'l : l ≤ (𝓝 0).coprod (𝓝 0)) :
Tendsto (fun x : M × M ↦ x.1 * x.2) l (𝓝 0) := by
simpa [inf_eq_right.mpr h'l] using tendsto_mul_coprod_nhds_zero_inf_of_disjoint_cocompact hl | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Algebra.Group.Submonoid.Basic",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Monoid.lean | tendsto_mul_nhds_zero_of_disjoint_cocompact | Let `M` be a topological space with a continuous multiplication operation and a `0`.
Let `l` be a filter on `M × M` which is both disjoint from the cocompact filter and less than or
equal to `(𝓝 0).coprod (𝓝 0)`. Then the multiplication map `M × M → M` tends to zero on `l`. |
Tendsto.tendsto_mul_zero_of_disjoint_cocompact_right {f g : α → M} {l : Filter α}
(hf : Tendsto f l (𝓝 0)) (hg : Disjoint (map g l) (cocompact M)) :
Tendsto (fun x ↦ f x * g x) l (𝓝 0) :=
tendsto_mul_nhds_zero_prod_of_disjoint_cocompact hg |>.comp (hf.prodMk tendsto_map) | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Algebra.Group.Submonoid.Basic",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Monoid.lean | Tendsto.tendsto_mul_zero_of_disjoint_cocompact_right | Let `M` be a topological space with a continuous multiplication operation and a `0`.
Let `f : α → M` and `g : α → M` be functions. If `f` tends to zero on a filter `l`
and the image of `l` under `g` is disjoint from the cocompact filter on `M`, then
`fun x : α ↦ f x * g x` also tends to zero on `l`. |
Tendsto.tendsto_mul_zero_of_disjoint_cocompact_left {f g : α → M} {l : Filter α}
(hf : Disjoint (map f l) (cocompact M)) (hg : Tendsto g l (𝓝 0)) :
Tendsto (fun x ↦ f x * g x) l (𝓝 0) :=
tendsto_mul_prod_nhds_zero_of_disjoint_cocompact hf |>.comp (tendsto_map.prodMk hg) | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Algebra.Group.Submonoid.Basic",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Monoid.lean | Tendsto.tendsto_mul_zero_of_disjoint_cocompact_left | Let `M` be a topological space with a continuous multiplication operation and a `0`.
Let `f : α → M` and `g : α → M` be functions. If `g` tends to zero on a filter `l`
and the image of `l` under `f` is disjoint from the cocompact filter on `M`, then
`fun x : α ↦ f x * g x` also tends to zero on `l`. |
tendsto_mul_cocompact_nhds_zero [TopologicalSpace α] [TopologicalSpace β]
{f : α → M} {g : β → M} (f_cont : Continuous f) (g_cont : Continuous g)
(hf : Tendsto f (cocompact α) (𝓝 0)) (hg : Tendsto g (cocompact β) (𝓝 0)) :
Tendsto (fun i : α × β ↦ f i.1 * g i.2) (cocompact (α × β)) (𝓝 0) := by
set l : Filter (M × M) := map (Prod.map f g) (cocompact (α × β)) with l_def
set K : Set (M × M) := (insert 0 (range f)) ×ˢ (insert 0 (range g))
have K_compact : IsCompact K := .prod (hf.isCompact_insert_range_of_cocompact f_cont)
(hg.isCompact_insert_range_of_cocompact g_cont)
have K_mem_l : K ∈ l := eventually_map.mpr <| .of_forall fun ⟨x, y⟩ ↦
⟨mem_insert_of_mem _ (mem_range_self _), mem_insert_of_mem _ (mem_range_self _)⟩
have l_compact : Disjoint l (cocompact (M × M)) := by
rw [disjoint_cocompact_right]
exact ⟨K, K_mem_l, K_compact⟩
have l_le_coprod : l ≤ (𝓝 0).coprod (𝓝 0) := by
rw [l_def, ← coprod_cocompact]
exact hf.prodMap_coprod hg
exact tendsto_mul_nhds_zero_of_disjoint_cocompact l_compact l_le_coprod |>.comp tendsto_map | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Algebra.Group.Submonoid.Basic",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Monoid.lean | tendsto_mul_cocompact_nhds_zero | If `f : α → M` and `g : β → M` are continuous and both tend to zero on the cocompact filter,
then `fun i : α × β ↦ f i.1 * g i.2` also tends to zero on the cocompact filter. |
tendsto_mul_cofinite_nhds_zero {f : α → M} {g : β → M}
(hf : Tendsto f cofinite (𝓝 0)) (hg : Tendsto g cofinite (𝓝 0)) :
Tendsto (fun i : α × β ↦ f i.1 * g i.2) cofinite (𝓝 0) := by
letI : TopologicalSpace α := ⊥
haveI : DiscreteTopology α := discreteTopology_bot α
letI : TopologicalSpace β := ⊥
haveI : DiscreteTopology β := discreteTopology_bot β
rw [← cocompact_eq_cofinite] at *
exact tendsto_mul_cocompact_nhds_zero
continuous_of_discreteTopology continuous_of_discreteTopology hf hg | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Algebra.Group.Submonoid.Basic",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Monoid.lean | tendsto_mul_cofinite_nhds_zero | If `f : α → M` and `g : β → M` both tend to zero on the cofinite filter, then so does
`fun i : α × β ↦ f i.1 * g i.2`. |
GroupWithZero.isOpen_singleton_zero [GroupWithZero M] [TopologicalSpace M]
[ContinuousMul M] [CompactSpace M] [T1Space M] :
IsOpen {(0 : M)} := by
obtain ⟨U, hU, h0U, h1U⟩ := t1Space_iff_exists_open.mp ‹_› zero_ne_one
obtain ⟨W, hW, hW'⟩ := exists_mem_nhds_zero_mul_subset isCompact_univ (hU.mem_nhds h0U)
by_cases H : ∃ x ≠ 0, x ∈ W
· obtain ⟨x, hx, hxW⟩ := H
cases h1U (hW' (by simpa [hx] using Set.mul_mem_mul (Set.mem_univ x⁻¹) hxW))
· obtain rfl : W = {0} := subset_antisymm
(by simpa [not_imp_not] using H) (by simpa using mem_of_mem_nhds hW)
simpa [isOpen_iff_mem_nhds] | lemma | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Algebra.Group.Submonoid.Basic",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Monoid.lean | GroupWithZero.isOpen_singleton_zero | null |
@[to_additive exists_open_nhds_zero_half]
exists_open_nhds_one_split {s : Set M} (hs : s ∈ 𝓝 (1 : M)) :
∃ V : Set M, IsOpen V ∧ (1 : M) ∈ V ∧ ∀ v ∈ V, ∀ w ∈ V, v * w ∈ s := by
have : (fun a : M × M => a.1 * a.2) ⁻¹' s ∈ 𝓝 ((1, 1) : M × M) :=
tendsto_mul (by simpa only [one_mul] using hs)
simpa only [prod_subset_iff] using exists_nhds_square this
@[to_additive exists_nhds_zero_half] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Algebra.Group.Submonoid.Basic",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Monoid.lean | exists_open_nhds_one_split | null |
exists_nhds_one_split {s : Set M} (hs : s ∈ 𝓝 (1 : M)) :
∃ V ∈ 𝓝 (1 : M), ∀ v ∈ V, ∀ w ∈ V, v * w ∈ s :=
let ⟨V, Vo, V1, hV⟩ := exists_open_nhds_one_split hs
⟨V, IsOpen.mem_nhds Vo V1, hV⟩ | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Algebra.Group.Submonoid.Basic",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Monoid.lean | exists_nhds_one_split | null |
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